Interpretation (logic) - Interpretations For Propositional Logic

Interpretations For Propositional Logic

The formal language for propositional logic consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, and propositional variables) and logical connectives. The only non-logical symbols in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed.

The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the truth values true and false. This function is known as a truth assignment or valuation function. In many presentations, it is literally a truth value that is assigned, but some presentations assign truthbearers instead.

For a language with n distinct propositional variables there are 2n distinct possible interpretations. For any particular variable a, for example, there are 21=2 possible interpretations: 1) a is assigned T, or 2) a is assigned F. For the pair a, b there are 22=4 possible interpretations: 1) both are assigned T, 2) both are assigned F, 3) a is assigned T and b is assigned F, or 4) a is assigned F and b is assigned T.

Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above.